|
%I #92 Apr 27 2023 17:53:45
%S 1,1,1,1,2,1,1,3,3,1,1,4,5,4,1,1,5,9,9,5,1,1,6,7,16,7,6,1,1,7,15,25,
%T 25,15,7,1,1,8,11,36,11,36,11,8,1,1,9,27,49,35,35,49,27,9,1,1,10,25,
%U 64,13,90,13,64,25,10,1,1,11,21,81,125,77,77,125,81
%N Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.
%C For any number n > 0, let f(n) be the polynomial in a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
%C This table has many similarities with A248601.
%C For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
%C Also, f(1) = 0 and f(2) = 1.
%C The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.
%C See A297473 for the main diagonal of T.
%C As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication (A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - _Peter Munn_, Mar 25 2020
%C From _Peter Munn_, Jun 16 2021: (Start)
%C The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.
%C Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.
%C There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.
%C There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively.
%C (End)
%H Rémy Sigrist, <a href="/A297845/b297845.txt">Table of n, a(n) for n = 1..5050</a>
%H Encyclopedia of Mathematics, <a href="https://encyclopediaofmath.org/wiki/Additive_arithmetic_function">Additive arithmetic function</a>
%H Encyclopedia of Mathematics, <a href="https://encyclopediaofmath.org/wiki/Isomorphism">Isomorphism</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Distributive.html">Distributive</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ring.html">Ring</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polynomial_ring">Polynomial ring</a>
%F T is completely multiplicative in both parameters:
%F - for any n > 0
%F - and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
%F - T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n + i - 1)^e_i.
%F For any m > 0, n > 0 and k > 0:
%F - T(n, k) = T(k, n) (T is commutative),
%F - T(m, T(n, k)) = T(T(m, n), k) (T is associative),
%F - T(n, 1) = 1 (1 is an absorbing element for T),
%F - T(n, 2) = n (2 is an identity element for T),
%F - T(n, 2^i) = n^i for any i >= 0,
%F - T(n, 4) = n^2 (A000290),
%F - T(n, 8) = n^3 (A000578),
%F - T(n, 3) = A003961(n),
%F - T(n, 3^i) = A003961(n)^i for any i >= 0,
%F - T(n, 6) = A191002(n),
%F - A001221(T(n, k)) <= A001221(n) * A001221(k),
%F - A001222(T(n, k)) = A001222(n) * A001222(k),
%F - A055396(T(n, k)) = A055396(n) + A055396(k) - 1 when n > 1 and k > 1,
%F - A061395(T(n, k)) = A061395(n) + A061395(k) - 1 when n > 1 and k > 1,
%F - T(A000040(n), A000040(k)) = A000040(n + k - 1),
%F - T(A000040(n)^i, A000040(k)^j) = A000040(n + k - 1)^(i * j) for any i >= 0 and j >= 0.
%F From _Peter Munn_, Mar 13 2020 and Apr 20 2021: (Start)
%F T(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).
%F T(n, m*k) = T(n, m) * T(n, k); T(n*m, k) = T(n, k) * T(m, k) (T distributes over multiplication).
%F A104244(m, T(n, k)) = A104244(m, n) * A104244(m, k).
%F For example, for m = 2, the above formula is equivalent to A048675(T(n, k)) = A048675(n) * A048675(k).
%F A195017(T(n, k)) = A195017(n) * A195017(k).
%F A248663(T(n, k)) = A048720(A248663(n), A248663(k)).
%F T(n, k) = A306697(n, k) if and only if T(n, k) = A329329(n, k).
%F A007913(T(n, k)) = A007913(T(A007913(n), A007913(k))) = A007913(A329329(n, k)).
%F (End)
%e Array T(n, k) begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10
%e ---+------------------------------------------------
%e 1| 1 1 1 1 1 1 1 1 1 1 -> A000012
%e 2| 1 2 3 4 5 6 7 8 9 10 -> A000027
%e 3| 1 3 5 9 7 15 11 27 25 21 -> A003961
%e 4| 1 4 9 16 25 36 49 64 81 100 -> A000290
%e 5| 1 5 7 25 11 35 13 125 49 55 -> A357852
%e 6| 1 6 15 36 35 90 77 216 225 210 -> A191002
%e 7| 1 7 11 49 13 77 17 343 121 91
%e 8| 1 8 27 64 125 216 343 512 729 1000 -> A000578
%e 9| 1 9 25 81 49 225 121 729 625 441
%e 10| 1 10 21 100 55 210 91 1000 441 550
%e From _Peter Munn_, Jun 24 2021: (Start)
%e The encoding, n, of polynomials, f(n), that is used for the table is further described in A206284. Examples of encoded polynomials:
%e n f(n) n f(n)
%e 1 0 16 4
%e 2 1 17 x^6
%e 3 x 21 x^3 + x
%e 4 2 25 2x^2
%e 5 x^2 27 3x
%e 6 x + 1 35 x^3 + x^2
%e 7 x^3 36 2x + 2
%e 8 3 49 2x^3
%e 9 2x 55 x^4 + x^2
%e 10 x^2 + 1 64 6
%e 11 x^4 77 x^4 + x^3
%e 12 x + 2 81 4x
%e 13 x^5 90 x^2 + 2x + 1
%e 15 x^2 + x 91 x^5 + x^3
%e (End)
%o (PARI) T(n,k) = my (f=factor(n), p=apply(primepi, f[, 1]~), g=factor(k), q=apply(primepi, g[, 1]~)); prod (i=1, #p, prod(j=1, #q, prime(p[i]+q[j]-1)^(f[i, 2]*g[j, 2])))
%Y Cf. A001221, A007913, A048720, A061395, A055396, A206284, A248601, A248663.
%Y Row n: n=1: A000012, n=2: A000027, n=3: A003961, n=4: A000290, n=5: A357852, n=6: A191002, n=8: A000578.
%Y Main diagonal: A297473.
%Y Functions f satisfying f(T(n,k)) = f(n) * f(k): A001222, A048675 (and similarly, other rows of A104244), A195017.
%Y Powers of k: k=3: A000040, k=4: A001146, k=5: A031368, k=6: A007188 (see also A066117), k=7: A031377, k=8: A023365, k=9: main diagonal of A329050.
%Y Integers in the ideal of the related ring (see Jun 2021 comment) generated by S: S={3}: A005408, S={4}: A000290\{0}, S={4,3}: A003159, S={5}: A007310, S={5,4}: A339690, S={6}: A325698, S={6,4}: A028260, S={7}: A007775, S={8}: A000578\{0}, S={8,3}: A191257, S={8,6}: A332820, S={9}: A016754, S={10,4}: A340784, S={11}: A008364, S={12,8}: A145784, S={13}: A008365, S={15,4}: A345452, S={15,9}: A046337, S={16}: A000583\{0}, S={17}: A008366.
%Y Equivalent sequence for polynomial composition: A326376.
%Y Related binary operations: A003991, A306697/A059896, A329329/A059897.
%K nonn,tabl,mult
%O 1,5
%A _Rémy Sigrist_, Jan 10 2018
%E New name from _Peter Munn_, Jul 17 2021
|
